Find the derivative of f(x) = ln(48e^3x − 11)
To find the derivative of f(x) = ln(48e^3x − 11), we can use the chain rule.
Step 1: Identify the inner function and the outer function.
In this case, the outer function is the natural logarithm (ln) and the inner function is 48e^3x − 11.
Step 2: Differentiate the outer function.
The derivative of ln(x) is 1/x.
Step 3: Differentiate the inner function.
To differentiate the inner function, we need to use the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), the derivative is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In this case, the inner function is 48e^3x − 11. Let's differentiate it:
d/dx (48e^3x − 11) = 48 * d/dx (e^3x) = 48 * 3e^3x = 144e^3x
Step 4: Combine the results from Steps 2 and 3.
The derivative of f(x) = ln(48e^3x − 11) is:
d/dx [ln(48e^3x − 11)] = 1/(48e^3x − 11) * 144e^3x = 144e^3x / (48e^3x − 11)
So, the derivative of f(x) = ln(48e^3x − 11) is 144e^3x / (48e^3x − 11).